To find out how much Derek needs in his retirement account on his 65th birthday, we will calculate the present value of a series of withdrawals he plans to make from age 66 to age 90, which is a total of 25 withdrawals (from age 66 to 90 inclusive).
The formula to calculate the present value of an annuity (where payments are made at regular intervals) is:
\[ PV = P \times \left(1 - (1 + r)^{-n}\right) / r \]
Where:
- \( PV \) = Present Value (the amount Derek needs at age 65)
- \( P \) = Annual payment (withdrawal amount)
- \( r \) = Interest rate per period
- \( n \) = Total number of payments
Given:
- \( P = 130817.00 \)
- \( r = 0.05 \)
- \( n = 25 \)
Now, we can plug in the values:
- Calculate \( (1 + r)^{-n} \):
\[ (1 + 0.05)^{-25} \approx (1.05)^{-25} \approx 0.295302 \]
- Now calculate \( 1 - (1 + r)^{-n} \):
\[ 1 - 0.295302 \approx 0.704698 \]
- Substitute back into the formula:
\[ PV = 130817 \times \left(0.704698 / 0.05\right) \]
\[ PV \approx 130817 \times 14.09396 \approx 1847102.05 \]
Therefore, Derek will need approximately $1,847,102.05 in his retirement account on his 65th birthday to withdraw $130,817.00 per year from age 66 to 90 at an interest rate of 5.00%.
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